2007/03/05 by Brent Doran, Frances Kirwan, Doran, Brent +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.math/0703131
openalex publication_date 2007/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points Xs and semistable points Xss, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of Xs, and (3) the existence of a canonical "enveloping quotient" variety of Xss, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X]H) when k[X]H is finitely generated.